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Theorem f1ocnvd 7376
Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
f1od.1 𝐹 = (𝑥𝐴𝐶)
f1od.2 ((𝜑𝑥𝐴) → 𝐶𝑊)
f1od.3 ((𝜑𝑦𝐵) → 𝐷𝑋)
f1od.4 (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
Assertion
Ref Expression
f1ocnvd (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)   𝐹(𝑥,𝑦)   𝑊(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem f1ocnvd
StepHypRef Expression
1 f1od.2 . . . . 5 ((𝜑𝑥𝐴) → 𝐶𝑊)
21ralrimiva 3149 . . . 4 (𝜑 → ∀𝑥𝐴 𝐶𝑊)
3 f1od.1 . . . . 5 𝐹 = (𝑥𝐴𝐶)
43fnmpt 6460 . . . 4 (∀𝑥𝐴 𝐶𝑊𝐹 Fn 𝐴)
52, 4syl 17 . . 3 (𝜑𝐹 Fn 𝐴)
6 f1od.3 . . . . . 6 ((𝜑𝑦𝐵) → 𝐷𝑋)
76ralrimiva 3149 . . . . 5 (𝜑 → ∀𝑦𝐵 𝐷𝑋)
8 eqid 2798 . . . . . 6 (𝑦𝐵𝐷) = (𝑦𝐵𝐷)
98fnmpt 6460 . . . . 5 (∀𝑦𝐵 𝐷𝑋 → (𝑦𝐵𝐷) Fn 𝐵)
107, 9syl 17 . . . 4 (𝜑 → (𝑦𝐵𝐷) Fn 𝐵)
11 f1od.4 . . . . . . 7 (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
1211opabbidv 5096 . . . . . 6 (𝜑 → {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = 𝐷)})
13 df-mpt 5111 . . . . . . . . 9 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
143, 13eqtri 2821 . . . . . . . 8 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
1514cnveqi 5709 . . . . . . 7 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
16 cnvopab 5964 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
1715, 16eqtri 2821 . . . . . 6 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
18 df-mpt 5111 . . . . . 6 (𝑦𝐵𝐷) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = 𝐷)}
1912, 17, 183eqtr4g 2858 . . . . 5 (𝜑𝐹 = (𝑦𝐵𝐷))
2019fneq1d 6416 . . . 4 (𝜑 → (𝐹 Fn 𝐵 ↔ (𝑦𝐵𝐷) Fn 𝐵))
2110, 20mpbird 260 . . 3 (𝜑𝐹 Fn 𝐵)
22 dff1o4 6598 . . 3 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
235, 21, 22sylanbrc 586 . 2 (𝜑𝐹:𝐴1-1-onto𝐵)
2423, 19jca 515 1 (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  wral 3106  {copab 5092  cmpt 5110  ccnv 5518   Fn wfn 6319  1-1-ontowf1o 6323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331
This theorem is referenced by:  f1od  7377  f1ocnv2d  7378  pw2f1ocnv  39976
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